By Kazez W. (ed.)
This can be half 2 of a two-part quantity reflecting the court cases of the 1993 Georgia overseas Topology convention held on the college of Georgia in the course of the month of August. The texts comprise learn and expository articles and challenge units. The convention coated a wide selection of subject matters in geometric topology. positive aspects: Kirby's challenge checklist, which includes an intensive description of the development made on all of the difficulties and incorporates a very entire bibliography, makes the paintings necessary for experts and non-specialists who are looking to know about the development made in many parts of topology. This record may possibly function a reference paintings for many years to return. Gabai's challenge checklist, which makes a speciality of foliations and laminations of 3-manifolds, collects for the 1st time in a single paper definitions, effects, and difficulties that can function a defining resource within the topic quarter.
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Extra resources for Geometric topology, Volume 2, Part 2
Remarks: (L. Taylor) The conjecture is true for fibered knots (even in homology spheres); however, the conjecture fails in general for knots in homology spheres. Perhaps it would be easier to show that the Seifert surface of K union the 2-handle is incompressible in ∂MK . 16). Update: (A) Still open. (B) The conjecture is true [355,Gabai,1987a,J. ]. In fact, Gabai shows that the Gromov norm and the singular Thurston norm of the generator of H2 (∂MK ) are both linear functions of genus K. 42 (Y.
Let f : S 1 × B 2 → S 3 be a trivialization of the normal disk bundle of K = f (S 1 × 0), for which f(S 1 × (1, 0)) lies on a Seifert surface for K (equivalently, f(S 1 × (1, 0)) represents 0 ∈ H1 (S 3 − K; Z)). This trivialization is called the 0-framing of K, and it defines the longitude λ (= f (S 1 ×(1, 0))) (the meridian µ is just f (point ×∂B 2)). Framing n is obtained from n ∈ π1 (SO(2)) (in this case, f (S 1 × (1, 0)) should wind n times around K as in a righthanded screw). If a 2-handle is added to B 4 along K with framing n, then the boundary is the result of n-surgery on S 3 along K.
16 CHAPTER 1. 14 (J. Simon) Characterize those knots K in S 3 for which the commutator subgroup G of π1(S 3 − K) has infinite weight (is not normally generated by a finite number of elements). Conjecture: K has infinite weight if K has a companion of winding number zero. Remarks: The conjecture is true for the untwisted double of any knot. Construct a knot K by putting a knot J in a solid torus T and tying T in a knot J ; then J is called a companion of K and J is a satellite of K. K represents an integer in H1 (T ; Z) which is called the winding number of J .
Geometric topology, Volume 2, Part 2 by Kazez W. (ed.)